《Linear Algebra Done Wrong》：为高阶学生打造的严谨入门指南

需积分: 49 167 浏览量更新于2024-07-17 11 收藏 1.45MB PDF 举报

"《线性代数做错了什么》（Linear Algebra Done Wrong）是一本由塞尔日·特雷尔（Sergei Treil）撰写，针对数学高级学生设计的教材。这本书并非以传统的、正统的方式来教授线性代数，而是以一种非典型的方式呈现，旨在提供给那些对抽象推理有所了解但希望深入学习严谨数学方法的学生，特别是那些希望在计算密集型课程中接触到更多理论和证明的学生。该书并非面向初级学习者，而是作为一门荣誉线性代数课程的讲义，它将线性代数的基础概念与严格的数学证明、形式定义相结合，让学生体验现代理论（抽象）数学的风格。与一般的入门线性代数教材相比，这本书更侧重于融合基本思想与具体实例，同时引入抽象的定义和构造，使得理论框架更加清晰且富有挑战性。作者通过这种方式，"错误地"引导读者去探索线性代数的深层次结构，鼓励他们跳出常规思维，学会如何从底层逻辑上理解和构建线性代数理论。这种“错误”的教学方法可能会让初学者感到困惑，但实际上是为了培养他们批判性思考和独立解决问题的能力，是为未来的数学研究和专业发展打下坚实基础的重要步骤。对于那些渴望深入理解线性代数本质，不满足于仅停留在公式和定理表面的读者来说，《线性代数做错了什么》是一本值得挑战自我的独特教材。它不仅涵盖了线性代数的基本内容，还提供了丰富的练习题和习题解答，帮助学生在实践中巩固所学知识，体验数学的理论美感和严谨逻辑。尽管标题带有争议性，但这本书对于寻求深化数学理解的读者来说，无疑是一次宝贵的学习之旅。"

6 1. Basic Notions

2. Linear combinations, bases.

Let V be a vector space, and let v

, v

, . . . , v

∈ V be a collection of vectors.

A linear combination of vectors v

, v

, . . . , v

is a sum of form

+ α

+ . . . + α

k=1

Deﬁnition 2.1. A system of vectors v

, v

, . . . v

∈ V is called a basis (for

the vector space V ) if any vector v ∈ V admits a unique representation as

a linear combination

v = α

+ α

+ . . . + α

k=1

The coeﬃcients α

, α

, . . . , α

are called coordinates of the vector v (in the

basis, or with respect to the basis v

, v

, . . . , v

Another way to say that v

, v

, . . . , v

is a basis is to say that for any

possible choice of the right side v, the equation x

+. . .+x

= v

(with unknowns x

) has a unique solution.

Before discussing any properties of bases

, let us give few a examples,

showing that such objects exist, and that it makes sense to study them.

Example 2.2. In the ﬁrst example the space V is R

. Consider vectors













, e













, e













, . . . , e













(the vector e

has all entries 0 except the entry number k, which is 1). The

system of vectors e

, e

, . . . , e

is a basis in R

. Indeed, any vector

v =













∈ R

can be represented as the linear combination

v = x

+ x

+ . . . x

k=1

and this representation is unique. The system e

, e

, . . . , e

∈ R

is called

the standard basis in R

the plural for the “basis” is bases, the same as the plural for “base”

2. Linear combinations, bases. 7

Example 2.3. In this example the space is the space P

of the polynomials

of degree at most n. Consider vectors (polynomials) e

, e

, . . . , e

∈ P

deﬁned by

:= 1, e

:= t, e

:= t

, e

:= t

, . . . , e

:= t

Clearly, any polynomial p, p(t) = a

t+a

+. . .+a

admits a unique

representation

p = a

+ a

+ . . . + a

So the system e

, e

, . . . , e

∈ P

is a basis in P

. We will call it the

standard basis in P

Remark 2.4. If a vector space V has a basis v

, v

, . . . , v

, then any vector

v is uniquely deﬁned by its coeﬃcients in the decomposition v =

k=1

. This is a very im-

portant remark, that

will be used through-

out the book. It al-

lows us to translate

any statement about

the standard column

space R

(or, more

generally F

) to a

vector space V with

a basis v

, v

, . . . , v

So, if we stack the coeﬃcients α

in a column, we can operate with them

as if they were column vectors, i.e. as with elements of R

(or F

if V is a

vector space over a ﬁeld F; most important cases are F = R of F = C, but

this also works for general ﬁelds F).

Namely, if v =

k=1

and w =

k=1

, then

v + w =

k=1

(α

+ β

i.e. to get the column of coordinates of the sum one just need to add the

columns of coordinates of the summands. Similarly, to get the coordinates

of αv we need simply to multiply the column of coordinates of v by α.

2.1. Generating and linearly independent systems. The deﬁnition

of a basis says that any vector admits a unique representation as a linear

combination. This statement is in fact two statements, namely that the rep-

resentation exists and that it is unique. Let us analyze these two statements

separately.

If we only consider the existence we get the following notion

Deﬁnition 2.5. A system of vectors v

, v

, . . . , v

∈ V is called a generating

system (also a spanning system, or a complete system) in V if any vector

v ∈ V admits representation as a linear combination

v = α

+ α

+ . . . + α

k=1

The only diﬀerence from the deﬁnition of a basis is that we do not assume

that the representation above is unique.

8 1. Basic Notions

The words generating, spanning and complete here are synonyms. I per-

sonally prefer the term complete, because of my operator theory background.

Generating and spanning are more often used in linear algebra textbooks.

Clearly, any basis is a generating (complete) system. Also, if we have a

basis, say v

, v

, . . . , v

, and we add to it several vectors, say v

n+1

, . . . , v

then the new system will be a generating (complete) system. Indeed, we can

represent any vector as a linear combination of the vectors v

, v

, . . . , v

and just ignore the new ones (by putting corresponding coeﬃcients α

= 0).

Now, let us turn our attention to the uniqueness. We do not want to

worry about existence, so let us consider the zero vector 0, which always

admits a representation as a linear combination.

Deﬁnition. A linear combination α

+ α

+ . . . + α

is called trivial

if α

= 0 ∀k.

A trivial linear combination is always (for all choices of vectors

, v

, . . . , v

) equal to 0, and that is probably the reason for the name.

Deﬁnition. A system of vectors v

, v

, . . . , v

∈ V is called linearly inde-

pendent if only the trivial linear combination (

k=1

with α

= 0 ∀k)

of vectors v

, v

, . . . , v

equals 0.

In other words, the system v

, v

, . . . , v

is linearly independent iﬀ the

equation x

+ x

+ . . . + x

= 0 (with unknowns x

) has only trivial

solution x

= x

= . . . = x

= 0.

If a system is not linearly independent, it is called linearly dependent.

By negating the deﬁnition of linear independence, we get the following

Deﬁnition. A system of vectors v

, v

, . . . , v

is called linearly dependent

if 0 can be represented as a nontrivial linear combination, 0 =

k=1

Non-trivial here means that at least one of the coeﬃcient α

is non-zero.

This can be (and usually is) written as

k=1

|α

| 6= 0.

So, restating the deﬁnition we can say, that a system is linearly depen-

dent if and only if there exist scalars α

, α

, . . . , α

k=1

|α

| 6= 0 such

that

k=1

= 0.

An alternative deﬁnition (in terms of equations) is that a system v

, . . . , v

is linearly dependent iﬀ the equation

+ x

+ . . . + x

= 0

(with unknowns x

) has a non-trivial solution. Non-trivial, once again again

means that at least one of x

is diﬀerent from 0, and it can be written as

k=1

| 6= 0.

10 1. Basic Notions

Proof. We already know that a basis is always linearly independent and

complete, so in one direction the proposition is already proved.

Let us prove the other direction. Suppose a system v

, v

, . . . , v

is lin-

early independent and complete. Take an arbitrary vector v ∈ V . Since the

system v

, v

, . . . , v

is linearly complete (generating), v can be represented

v = α

+ α

+ . . . + α

k=1

We only need to show that this representation is unique.

Suppose v admits another representation

v =

k=1

eα

Then

k=1

(α

− eα

k=1

−

k=1

eα

= v − v = 0.

Since the system is linearly independent, α

− eα

= 0 ∀k, and thus the

representation v = α

+ α

+ . . . + α

is unique. 

Remark. In many textbooks a basis is deﬁned as a complete and linearly

independent system (by Proposition 2.7 this deﬁnition is equivalent to ours).

Although this deﬁnition is more common than one presented in this text, I

prefer the latter. It emphasizes the main property of a basis, namely that

any vector admits a unique representation as a linear combination.

Proposition 2.8. Any (ﬁnite) generating system contains a basis.

Proof. Suppose v

, v

, . . . , v

∈ V is a generating (complete) set. If it is

linearly independent, it is a basis, and we are done.

Suppose it is not linearly independent, i.e. it is linearly dependent. Then

there exists a vector v

which can be represented as a linear combination of

the vectors v

, j 6= k.

Since v

can be represented as a linear combination of vectors v

, j 6= k,

any linear combination of vectors v

, v

, . . . , v

can be represented as a linear

combination of the same vectors without v

(i.e. the vectors v

, 1 ≤ j ≤ p,

j 6= k). So, if we delete the vector v

, the new system will still be a complete

one.

If the new system is linearly independent, we are done. If not, we repeat

the procedure.

Repeating this procedure ﬁnitely many times we arrive to a linearly

independent and complete system, because otherwise we delete all vectors

and end up with an empty set.

剩余275页未读，继续阅读

clf2017

粉丝: 0

《Linear Algebra Done Wrong》：为高阶学生打造的严谨入门指南

linear algebra done wrong

线性代数做错了Linear Algebra Done Wrong

Linear Algebra Done Right(Second Edition) By Sheldon Axler

线性代数资料Linear Algebra

"Wrong usage of OpenCV's Eigen utility header. Include Eigen's headers first.

Linear algebra done right

Linear Algebra Done Right

Introduction to Linear Algebra-5th edition-2016.pdf.zip

Introduction to Linear Algebra_5th_2016_by Gilbert Strang.pdf

Introduction to Linear Algebra(第四版答案).pdf

最新资源